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First let me define what is the "connective constant" of a two dimensional lattice. Let $c_{n}$ denote the number of $n$ step self-avoiding walks starting from a fixed origin point in the lattice. Then the limit (which can be shown to exist) $$ \mu := \lim_{n \rightarrow \infty } c_{n}^{\frac{1}{n}} $$ is the connective constant of the lattice.

My question is the following: does this definition also make sense on any periodic tiling of the plane? To clarify the difference, a lattice is a discrete subgroup of $\mathbb{R}^2$. So I think a lattice does give a tiling of the plane, but not the other way round.

$\textbf{Edit:}$ It seems from the comments I received that $\mu$ does make sense for any tiling. Is there any reason why $\mu$ has been studied only for a lattice and not for other tilings? Is there a reason why this is a "natural" question for lattices and an "artificial" question for other tilings.

Of course I could be wrong, i.e. this question has been studied for other lattices. In which case can someone point out any reference for this? Presently $\mu$ is known for the Hexagonal Lattice (Smirnov) and conjectured for a few other lattices. I am looking for conjectures/numerical simulations for tilings that are not lattices.

For simplicity I was talking about $2$ dimensions, but the question makes sense in any dimension.

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  • $\begingroup$ Could you describe explicitly an example of a tiling of $\mathbb{R}^2$ by $4g$-gon with $g>1$? (I assume the tiles should be convex and congruent, but I could be wrong.) $\endgroup$
    – Wlodek Kuperberg
    Commented Feb 22, 2014 at 15:36
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    $\begingroup$ I think I was mistaken; I can't really think of a tiling of R^2 even for an 8-gon. I do want the tiling to be congruent. $\endgroup$
    – Ritwik
    Commented Feb 22, 2014 at 17:44
  • $\begingroup$ Then the only tilings you can have are by polygons with at most six sides, and if you want the tiles to be parallel translates of each other (rotations not allowed), then triangles and pentagons are out. $\endgroup$
    – Wlodek Kuperberg
    Commented Feb 22, 2014 at 18:56
  • $\begingroup$ I'm not sure what you have in mind, but the question probably makes perfectly good sense for non-congruent tilings of the plane - whether periodic or not. I'm sure you could make a definition of $\mu$ that would work for any periodic tiling of the plane. I would be very confident that it would also work for aperiodic tilings such as the Penrose tiling also (without having any details in mind). $\endgroup$
    – Anthony Quas
    Commented Feb 22, 2014 at 19:10
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    $\begingroup$ Yes, your question is meaningful and natural. But it should not be too surprising if it has not been studied before - this happens quite often. A new discovery often leads to many new questions. I guess there are more such questions than all mathematicians in the world can handle or even think about. By the way, if you're interested in tilings, I recommend "Tilings and Patterns" (1987) - a great book by Grünbaum and Shephard. $\endgroup$
    – Wlodek Kuperberg
    Commented Feb 23, 2014 at 15:06

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